173ed49

Division of the just 49:1 interval into 173 equal parts yields a remarkable non-octave-equivalent equally tempered scale with step width very close to 31edo, yet it is much more suited to melodies and harmonies spanning more than one octave. The octaves in this scale are stretched by 7.3 cents, and the step width is about 0.23 cents wider than in 31edo (which has 174 equal steps in its tempered version of 49:1).

This scale and 18edf perform almost identically except over very large distances.

Out of all the harmonics between 1 and 49 and other low-harmonic entropy intervals within this range, this tuning matches the overwhelming majority with tolerable accuracy. The only harmonics that aren't matched well are 27, 33, and 37. This scale also has very good perfect fifths (within a cent of just intonation), although its fourths are not as good.

The 49:1 interval, 6737.6518 cents, could be called a "wide fortieth" since it consists of five octaves plus a 49:32 "wide fifth". This is more than half the average hearing range of a human, and thus actual instruments would find it seldom necessary to cover much more than this range. Extending too far beyond this range offers diminishing returns anyway, since harmonics above the 51st become very poorly matched compared to those below. Thus, this temperament is best used over a finite range (173 or perhaps 175 steps at the maximum).

The intervals within this scale are:

Interval	Width in steps	Width in cents	Approximations (error)
Diesis	1	38.946	49:48 45:44
Chromatic semitone	2	77.891	25:24 22:21
Diatonic semitone, secor	3	116.838	16:15 15:14
Neutral second	4	155.784	12:11
Whole tone	5	194.730	10:9 9:8
Septimal whole tone	6	233.676	8:7
Septimal minor third	7	272.622	7:6
Minor third	8	311.568	6:5
Neutral third	9	350.514	11:9
Major third	10	389.460	5:4
Septimal major third	11	428.406	9:7
Subfourth	12	467.351	21:16
Perfect fourth	13	506.298	4:3
Superfourth	14	545.243	11:8 15:11
Lesser septimal tritone	15	584.189	7:5
Greater tritone	16	623.135	10:7 13:9
Subfifth	17	662.081	22:15
Perfect fifth	18	701.027	3:2
Superfifth	19	739.973	20:13
Undecimal minor sixth	20	778.919	11:7
Minor sixth, golden ratio	21	817.865	8:5 φ:1
Undecimal neutral sixth	22	856.811	18:11
Major sixth	23	895.757	5:3
Septimal major sixth	24	934.703	12:7
Septimal minor seventh	25	973.649	7:4
Minor seventh	26	1012.595	9:5
Neutral seventh	27	1051.541	11:6
Major seventh	28	1090.487	15:8
Supermajor seventh	29	1129.433
Diminished octave	30	1168.379
(Stretched) octave	31	1207.325	2:1
Augmented octave	32	1246.271
Enneadecimal minor ninth	33	1285.217	19:10
Minor ninth	34	1324.163	15:7
Neutral ninth	35	1363.109	11:5
Major ninth	36	1402.055	9:4
Supermajor ninth	37	1441.001
Septimal minor tenth	38	1479.947	7:3
Minor tenth	39	1518.893	12:5
Neutral tenth	40	1557.839	22:9
Major tenth	41	1596.785	5:2
Septimal major tenth	42	1635.730	14:7
Sub-eleventh	43	1674.676	21:8
Perfect eleventh	44	1713.622	8:3
Tridecimal super-eleventh	45	1752.568	13:7
Lesser eka-tritone*	46	1791.514
Greater eka-tritone*	47	1830.460	23:8
Diminished twelfth	48	1869.406
Tritave; perfect twelfth	49	1908.352	3:1
Augmented twelfth	50	1947.298
Subminor thirteenth, pi	51	1986.244	22:7 π:1
Minor thirteenth	52	2025.190	16:5 13:4
Neutral-major thirteenth	53	2064.136	10:3 (flat)
Major thirteenth	54	2103.082	10:3 (sharp)
Supermajor thirteenth	55	2142.028
Septimal minor fourteenth	56	2180.974	7:2
Minor fourteenth	57	2219.920	18:5
Neutral fourteenth	58	2258.866	11:3
Major fourteenth	59	2297.819	15:4
Supermajor fourteenth	60	2336.758
Diminished double octave	61	2375.704
Double octave (fifteenth)	62	2414.650	4:1
Augmented double octave	63	2453.596	33:8
Minor sixteenth	64	2492.542	21:5 17:4
Neutral sixteenth	65	2531.488	13:3
Neutral-major sixteenth	66	2570.434	22:5
Major sixteenth	67	2609.380	9:2
Supermajor sixteenth	68	2648.326	14:3 (flat)
Minor seventeenth	69	2687.272	14:3 (sharp) 19:4
Neutral seventeenth	70	2726.217
Major seventeenth, 5th harmonic (narrow)	71	2765.153	5:1
Major seventeenth, 5th harmonic (wide)	72	2804.109	5:1
Supermajor seventeenth	73	2843.055	13:5
Eighteenth (narrow)	74	2882.001	21:4 16:3 (flat)
Eighteenth (wide)	75	2920.947	16:3 (sharp) 27:5
Augmented eighteenth	76	2959.893	11:2
Septendecimal dvi-tritone	77	2998.839	17:3
Greater dvi-tritone	78	3037.785
Diminished nineteenth	79	3076.731
Nineteenth; 6th harmonic	80	3115.677	6:1
Augmented nineteenth	81	3154.623
Minor twentieth	82	3183.569
Minor-neutral twentieth	83	3232.515	13:2
Major-neutral twentieth	84	3271.461	20:3
Major twentieth	85	3310.407	27:4
7th harmonic (narrow)	86	3349.353	7:1
7th harmonic (wide), subminor twenty-first	87	3388.299	7:1
Minor twenty-first	88	3427.245	29:4
Neutral twenty-first	89	3466.191	22:3 15:2 (flat)
Major twenty-first	90	3505.137	15:2 (sharp)
Supermajor twenty-first	91	3544.083
Triple octave; twenty-second; 8th harmonic (narrow)	92	3583.029	8:1
Triple octave; twenty-second; 8th harmonic (wide)	93	3621.975	8:1
Subminor twenty-third	94	3660.921	25:3
Minor twenty-third	95	3699.867	17:2
Minor-neutral twenty-third	96	3738.813	26:3
Major-neutral twenty-third	97	3777.759
Major twenty-third; 9th harmonic	98	3816.704	9:1
Subminor twenty-fourth	99	3855.650
Minor twenty-fourth	100	3894.597	19:2
Minor-neutral twenty-fourth	101	3933.543
Major-neutral twenty-fourth; 10th harmonic	102	3972.489	10:1
Major twenty-fourth	103	4011.434
Supermajor twenty-fourth; diminished twenty-fifth	104	4050.380	21:2 (flat)
Twenty-fifth	105	4089.326	21:2 (sharp) 32:3
Augmented twenty-fifth	106	4128.272
11th harmonic	107	4167.218	11:1
Tri-tritone	108	4206.164	34:3
Diminished twenty-sixth	109	4245.110	23:2
Twenty-sixth; 12th harmonic (flat)	110	4284.056	12:1
Twenty-sixth; 12th harmonic (sharp)	111	4323.002	12:1
Subminor twenty-seventh	112	4361.948	25:2
Minor twenty-seventh	113	4400.894
13th harmonic	114	4439.840	13:1
Major-neutral twenty-seventh	115	4478.786
Major twenty-seventh	116	4517:732	27:2
14th harmonic	117	4556.678	14:1
Minor twenty-eighth	118	4595.624
Minor-neutral twenty-eighth	119	4634.57	29:2
15th harmonic	120	4673.516	15:1
Major twenty-eighth	121	4712.462
Diminished quadruple octave	122	4751.408
Twenty-ninth; quadruple octave; 16th harmonic	123	4790.354	16:1
Augmented quadruple octave	124	4829.300
Subminor thirtieth	125	4868.246
Minor thirtieth; 17th harmonic	126	4907.191	17:1
Neutral thirtieth	127	4946.137
Major thirtieth, 18th harmonic (narrow)	128	4985.083	18:1
Major thirtieth, 18th harmonic (wide)	129	5024.029	18:1
Subminor thirty-first	130	5062.975	56:3
19th harmonic	131	5101.921	19:1
Neutral thirty-first	132	5140.867	39:2
20th harmonic	133	5179.813	20:1
Major thirty-first	134	5218.759	41:2
21st harmonic	135	5257.705	21:1
	136	5296.651	43:2
22nd harmonic	137	5335.597	22:1
	138	5374.543	45:2
23rd harmonic	139	5413.489	23:1
	140	5452.435	47:2
24th harmonic	141	5491.381	24:1
	142	5530.327	49:2
25th harmonic	143	5569.273	25:1
	144	5608.22	51:2
26th harmonic	145	5647.16	26:1
flat 27th harmonic	146	5686.11	27:1 (flat)
sharp 27th harmonic	147	5725.06	27:1 (sharp)
28th harmonic	148	5764.00	28:1
	149	5802.95
29th harmonic	150	5841.89	29:1
30th harmonic	151	5880.84	30:1
	152	5919.79	61:2
31st harmonic	153	5958.73	31:1
Quintuple octave, 32nd harmonic	154	5997.68	32:1
Flat 33rd harmonic	155	6036.62	33:1 (flat)
Sharp 33rd harmonic	156	6075.57	33:1 (sharp)
34th harmonic	157	6114.52	34:1
35th harmonic	158	6153.46	35:1
36th harmonic	159	6192.41	36:1
Flat 37th harmonic	160	6231.35	37:1 (flat)
Sharp 37th harmonic	161	6270.30	37:1 (flat)
38th harmonic	162	6309.25	38:1
39th harmonic	163	6348.19	39:1
40th harmonic	164	6387.14	40:1
41st harmonic	165	6426.08	41:1
42nd harmonic	166	6465.08	42:1
43rd harmonic	167	6503.98	43:1
44th harmonic	168	6542.92	44:1
45th harmonic	169	6581.87	45:1
46th harmonic	170	6620.81	46:1
47th harmonic	171	6659.76	47:1
48th harmonic	172	6698.71	48:1
49th harmonic	173	6737.65	49:1 ===(just)===
50th harmonic	174	6776.60	50:1
51st harmonic	175	6815.55	51:1

*An eka-tritone (named by analogy with the periodic table) is an octave plus a tritone. A dvi-tritone is two octaves plus a tritone.
**The 5:1, 7:1, 8:1, 12:1, and 18:1 intervals are split, yet all have a relatively high tolerance for mistuning, so in each case, both approximations are reasonable. When designing instruments to play in this tuning, it might be a good idea to dampen the 5th, 7th, 8th, and 12th harmonics while detuning the others slightly toward their corresponding scale degrees.